Microsoft Edge aka the Project Spartan browser is due to launch with Windows 10 later this summer. Post-release, it will receive support for third-party extensions later in the year. Developers can easily port over existing extensions from Firefox or Chrome, making Edge a very powerful option for many users.
The question lends itself to asking about Windows Phone. After all, Microsoft Edge is a universal Windows app, so extension support would seem highly likely. However, as it turns out, it is rather nuanced.